Co2MnSi MCPs were measured in a magnetic field of 1T at 300K on the BL08w beamline
at SPring-8. The [100], [110] and [111] directions were measured so multiple directions could be compared to the theoretical calculations. Alignments were performed through the use of back-scattered Laue diffraction, prior to the experiment.
The theoretical [110] profile can be seen in figure 6.10. The unconvoluted and convo- luted profiles have been plotted against one another to show how dramatic an effect convolution can have on the shape of a theoretical MCP. The usefulness of convolu- tion is in its attempt to mimic the resolution of the actual experimental data. Often (and is quite clearly the case here), the Umklapp features present in metallic MCPs are very pronounced in theoretical MCPs. Convolution often alludes to why the umklapp features are not so prominent in the experimental data.
Si 2 p D e n s i t y o f S t a t e s [ s t a t e s / e V ] E-E F [eV]
Figure 6.9: The calculated DOS of Co2MnSi together with the partial DOS of the
compositional atoms.
Figure 6.11 plots the experimental MCPs against the theoretical MCPs calculated. The areas of all profiles have been normalised to 1 for the purpose of comparison and the theoretical profiles have been convoluted using a gaussian of FWHM = 0.44 a.u. When studying multiple directions using MCS, it is beneficial to not only measure the direc- tion specific MCPs in order to study how the shapes change, but to also calculate the
0 1 2 3 4 5 6 7 8 9 10 0.0 0.1 0.2 0.3 0.4 ELK Co 2 MnSi [110] unconvoluted ELK Co 2
MnSi [110] convoluted (0.44 a.u)
J m a g ( p z ) [ a r b . ] p z [a.u]
Figure 6.10: Theoretical [110] convoluted and unconvoluted profiles. The unconvoluted profile shows the presence ofUmklappfeatures, typical in a material which has a metallic (or in this case, partially metallic) band structure. The convolution was performed using a gaussian of FWHM = 0.44a.u.
two directional MCPs. This has the effect of removing the less anisotropic, localised contributions to the MCP (higher momentum contributions). Since DFT calculations are quite apt in describing the more localised contributions to the MCP, theanisotropic
MCP gives a much morepureplot of the itinerant components of the MCP. Comparing the experimental and theoretical anisotropies may therefore be a better indicator of the calculation’s ability to model the itinerant region of the EMD than comparing the ex- perimental directional MCPs to the theory. As such, the experimental and theoretical anisotropies have been included in the figure. The agreement with the experimental data is very good, with the calculations capturing the broadness and the periodic Umklapp features. Even at low momentum, where contributions are due to the more itinerant electrons of the material, the agreement is very reasonable. While there are deviations present between the data and calculations, compared to the earlier experiments,24,122the
0.00 0.15 0.30 0.45 0.00 0.12 0.24 0.36 0 2 4 6 8 10 0.00 0.15 0.30 0.45 J m a g ( p z ) [ a r b . ] Co 2
MnSi SPring-8 Feb 2015 ELK p z [a.u] [100] [110] [111] [100] - [110] [110] - [111] [111] - [100] 0 2 4 6 8 -0.05 0.00 0.05 0 2 4 6 8 -0.05 0.00 0.05 0 2 4 6 8 -0.15 -0.10 -0.05 0.00 0.05
Figure 6.11: Experimental Co2MnSi along the [100], [110] and [111] directions plot-
ted against theoretical profiles calculated in ELK. The insets plot the experimental anisotropy against the theoretical anisotropy.
it can be seen that the agreement is again, excellent, with the oscillatory features and general shape being very well described by the DFT calculation.
6.4.1 Directional Analysis
For the [100] direction, the calculation captures the essential features of the experimental data very well. Unlike the other two directions, there is no low momentum dip in the experimental profile of the [100] direction, and this is captured in the calculation. The broadness is also replicated well and the Umklapp features seen in the theoretical profile can be observed in the experimental data at 1.5, 3 and 4 a.u.
Likewise, the theoretical [110] profile has very good agreement with the experimental data, matching the broadness of the data and replicating the slight low energy dip seen in the data. At this level of convolution, the Umklapp features cannot be seen much like the theoretical profile, the data resembles that of a smooth curve. Unlike the theoretical calculation however, the experimental data only has one peak at low momentum instead of two. The absence of this peak may be an anomaly due to the binning used to generate this data. The data points in this momentum region oscillate slightly indicating that with higher statistics, a feature may develop.
The [111] calculation captures the broadness of the data and replicates the low energy
dip well, but like the [110] direction, seems to over-estimate the peak in the profile at about 1.5 a.u. Contrary to the other two directions, the [111] direction has very large Umklapp features which do not seem to be observed in the experimental data.
6.4.2 Anisotropy Analysis
Turning attention to the anisotropies, the calculations very well reflect how the electron momentum distribution changes in momentum space. The general shapes of all the anisotropies are very well reproduced. As was the case with the MCPs, the anisotropies’ experimental periodicity seems to be shrouded possibly due to the experimental resolu- tion as in the [100]-[110] and [111]-[100] profiles, the subtle peak at 2 a.u is not replicated well by the data. For the [110]-[111] anisotropy, which has larger, broader periodic fea- tures, the periodicity and shape of the experimental anisotropy is very well reproduced by the calculation.
To summarise, the agreement between the experimental data and the GGA calculation is very impressive. Except for perhaps the [111] direction (due to its failure to adequately describe the shape of the experimental data), the calculation captures the momentum space distribution of Co2MnSi very well. To extend the work further, an investigation
into how electron correlation affects the EMD was performed in order to see how this will change the shapes and character of the MCPs.
6.4.3 Correlation effects study using GGA+U
For many compounds, the LDA or GGA exchange correlation functionals are sufficient for generalising the interactions (or lack of interactions) between the itinerant, valence electrons of the system. Turning off the Coulomb interaction and treating the electrons as a homogenous electron gas (LDA) and taking into consideration, the change in elec- tron density permeating thatsea (GGA) can produce very accurate results. For systems which are morecorrelated, where the Coulomb interactions are not only strong between different electrons, but also the electrons’ own self interaction, different approaches can be implemented to account for this correlation effect. Since variations in the lattice pa- rameter yield no discernible impact on the MCPs, investigating how strongly correlated the electrons are, and what affect this correlation has on the band structure of Co2MnSi
is very useful to test how rigorous Co2MnSi’s band structure is.
In the same vein as similar work from the past,123 the on-site electron correlation will be simulated using the GGA+U method.124 The following work investigates how the band structure and magnetic moments are affected with the application of U. For all calculations,J = 0.88 eV and theAround Mean Field (AMF) double counting scheme125 was employed. Calculations were performed whereU was applied to the Codelectrons, the Mn delectrons, and both Co and Mn delectrons. No calculations were performed whereU was applied to the Si as the Si 2p electrons are too low in energy to affect the magnetic properties and band gap of the compound.
UCo Calculations
Applying U to the Co d electrons results in a very subtle change to the electronic and magnetic structure of the system. Figure 6.12(a) shows how the spin moments of Co2MnSi with the application of U to the Co delectrons.
0 2 4 6 8 10 0 1 2 3 4 5 6 T o t a l M o m e n t [ B ] U Co [eV] 0 2 4 6 8 10 0 1 2 3 4 5 6 Co Moment Mn Moment Total Moment Mn/Co Ratio T o t a l M o m e n t [ B ] U Mn [eV] 0 20 40 60 80 100 D S P [ % ] 0 20 40 60 80 100 D S P [ % ]
Figure 6.12: Spin moment quantities and DSPs calculated for appliedUCo (a) andUMn
(b).
AsU is increased, a change in ratio between the Mn and Co spin moments is observed which is why the total moment remains at 5.00 µB. Since the band structure is shifted minimally for smallerU values, the MCPs reflect this with very small deviations from the base GGA result. AtU >6 eV, the change in Mn:Co ratio continues, increasing the total moment above 5.00µB, destroying the half-metallicity. It is concluded that correlation affects in the Co atoms play no role in changing the Co2MnSi model. However, it is
unlikely the correlation effects present in Co will be of such a magnitude and hence, are not considered to be of physical significance. To conclude, attempting to model Co’s
correlation results in minimal changes to the electronic structure of Co2MnSi.
UMn Calculations
For the effects ofU on the Mndelectrons, much like the Co results, for U <6 eV, the effects on the band structure and magnetic structure are minimal. Figure 6.12(b) plots the spin moments of Co2MnSi with the application of U to the Mn d electrons. Much
like with the applied Co U, forU <6 eV, the total moment remains at a constant 5.00
µB as the Mn:Co ratio shifts. Compared to the CoU results, the change in Mn and Co moments is larger creating a more significant change in the band structure. This peaks at U = 6 eV where the DSP drops significantly. As for higher values, the magnetic structure completely changes as the total moment drops to ∼ 1.01 µB. At this value, the DSP increases again as the total moment has approached an almost integer value. As evidenced from the magnetometry (§6.2.2) the total moment of Co2MnSi is not this
small and so the results forU > 6 eV can be interpreted as being non-physical. Much as is the case with Co, Mn’s correlation effects are not thought to be this high in energy.
UCo,Mn Calculations
In these sets of calculations, keeping theUCo:UMn ratio fixed to 1 : 1,U was increased
from 0 eV to 10 eV. Consistent with the previous two sets of calculations, the Mn:Co total moment ratio falls until at U = 6 eV where the half-metallicity is destroyed and the spin moment drops to 2.39µB.
Attention must be drawn to the fact that a 6×6 matrix of different calculations can be performed for the case whereUCo andUMnare both varied. Future work could attempt
to calculate these different elements of the matrix.
As a result of this correlation investigation, the theoretical work makes quite a substan- tial case for the lack of correlation effects in Co2MnSi.
6.4.4 Relationship between Site-Disorder and Magnetic Moment
Studying how the effects of disorder change the band structure (and thus, magnetic properties) may be important for understanding analysing the Co2MnSi data. Numerous
studies investigating the disorder in the compound have been performed using DFT since variations in the stoichiometry of the Heusler materials may lead to explanations as to why they are half-metallic (or equally, why they are not). Picozzi et al. investigated the four most likely types of disorder to occur in Co2MnSi. The following are ranked in
terms of theoretical formation energy ∆E from smallest to largest:126 1. Mn→ Co non-stoichiometric disorder
2. Co→ Mn non-stoichiometric disorder
3. Co↔ Mn stoichiometric disorder
4. Mn↔ Si stoichiometric disorder
Other types of disorder are energetically unfavourable and hence, were not studied. The results from FLAPW-GGA calculations suggested that the non-stoichiometric disorder introduced from Co atoms substituting Mn atoms drastically lowered the spin polarisa- tion, introducing Co states into the minority band atEF. Conversely, non-stoichiometric
disorder of Co atoms by Mn makes no impact on the minority states of Co2MnSi preserv-
ing the half-metallicity. Whilst the formation energies are reported as being different, experimentally, these types of disorder are equally prevalent and so, it is predicted that non-stoichiometric disordering of Co and Mn atoms will play a dominant role in the destruction of Co2MnSi’s half-metallicity. Galanakis et al. find in contradiction with
Picozziet al.changes to the spin polarisation of Co2MnSi when stoichiometric Mn↔Si
disorder is modelled.127 Even small concentrations of ∼5% cause changes to the total moment of the system, destroying the half-metallicity.
Finally, Pandeyet al. found (also using the CPA) that for very small disorder (changes of a fraction of 1 %), quite significant changes can occur to the total moment of the system.128 As such, being able to compare disorder effects to the experimental MCS and SQuID data will prove to be very useful in the analysis of this compound.